It is trivial to check that ω{\displaystyle \omega } is recursive, the successor of a recursive ordinal is recursive, and the set of all recursive ordinals is closed downwards. The supremum of all recursive ordinals is called the Church–Kleene ordinal and denoted by ω1CK{\displaystyle \omega _{1}^{CK}}. Indeed, an ordinal is recursive if and only if it is smaller than ω1CK{\displaystyle \omega _{1}^{CK}}. Since there are only countably many recursive relations, there are also only countably many recursive ordinals. Thus, ω1CK{\displaystyle \omega _{1}^{CK}} is countable.